PHYSICAL REVIEW E 88, 035201 (2013)

Finite-depth capillary-gravity dromions

Yong Liang and Mohammad-Reza AlamDepartment of Mechanical Engineering, University of California, Berkeley, California 94720, USA

(Received 16 March 2013; published 13 September 2013)

It is known that the governing equations for the evolution of the envelope of weakly nonlinear capillary-gravitywave packets, the so-called Davey-Stewartson equation, admit dromion solutions in the limit of shallow waterand strong surface tension, i.e., when kh � 1 and the Bond number (Bo) ≡ T/ρgh2 > 1/3 (where T is thesurface tension coefficient and h the water depth). Here we show that capillary-gravity dromions exist beyondthis limit for a broad range of finite water depths, i.e., for kh � O(1), as well as for subcritical Bond numbers,i.e., for Bo < 1/3.

DOI: 10.1103/PhysRevE.88.035201 PACS number(s): 47.35.Fg, 47.35.Pq, 47.35.Bb

In the context of nonlinear water wave theory, an intriguingquestion has always been in regard to whether fully localized3D wave structures, counterparts of 2D solitons, can exist.These structures are important because, if they exist, they cantransport mass, momentum, and energy over long distances[1]. For pure gravity waves this possibility is already ruledout [2,3], but a few limiting cases of capillary-gravity andflexural-gravity wave equations admit such solutions in theform of dromions and lumps (e.g., Refs. [4–9]).

Dromions are fully localized 3D surface structures thatcan travel with a constant speed without changing the form.They are formed at the intersection of line-solitary mean-current tracks (dromos means “tracks” in Greek) and haveexponentially decaying tails. Lumps, on the other hand,have a surface-elevation and mean-current with algebraicallydecaying tails. A number of three-dimensional systems areknown to admit lumps, including the Kadomtsev-Petviashviliequation [10,11], Benney-Luke equations [6], the Davey-Stewartson equation [7,12], for shallow and deep water andthe full Euler equation [13,14]. However physically relevantdromion solutions are limited so far to the shallow water andsurface tension–dominated regimes of the Davey-Stewartsonequation.

The Davey-Stewartson equation [15,16] is the three-dimensional extension of the nonlinear Schrödinger equation(NLS) [17,18] but differs significantly from the NLS in that theenvelope evolution is coupled with the mean field. The Davey-Stewartson equation was first derived for the finite-depthgravity waves [15,16] and then extended to include the effectof surface tension [19] and surface flexural rigidity [5,20],among several other applications in diverse areas of science(e.g., Refs. [21–25]). The Davey-Stewartson equation in thelimit of shallow water and strong surface tension (i.e., kh � 1and Bo > 1/3, the so-called DSI limit) is known to admitdromion solutions [4,26,27].

Here we show that dromions also exist in finite-depth waters[i.e., kh � O(1)] and for subcritical Bond numbers (i.e., Bo <1/3) for which the Davey-Stewartson equation is not integrable[28]. We present a hybrid numerical technique that utilizesrespectively a pseudospectral method for the elliptic operatorand a pseudotime integration for the hyperbolic operator ofthe governing equation. The scheme converges efficiently todromion solutions well beyond the range of classical DSI. Wefurther perform a comprehensive search for dromions over theentire parameter space of kh-Bo and show that such structures,

in fact, exist over a large subspace of the focusing elliptic-hyperbolic Davey-Stewartson equation.

Consider free propagating waves on the surface of anincompressible, inviscid, and homogeneous fluid with densityρ, depth h, and surface tension coefficient T . We define aCartesian coordinate system with the x and y axes on themean bottom and z-axis positive upward. Assuming that theflow is irrotational a potential function φ can be definedsuch that ∇φ ≡ �u, where �u = �u(x,y,z,t) is the Eulerianvelocity of the flow field [29]. If η = η(x,y,t) denotes theelevation of the free surface from the mean water level,then equations governing the evolution of disturbances in thisfield read

∇2φ = 0, 0 < z < h + η(x,y), (1a)ηt + ηxφx + ηyφy = φz, z = h + η(x,y), (1b)φt + gη + 1

2|∇φ|2 − T

ρ(w1,x + w2,y) = 0,

(1c)z = h + η(x,y),φz = 0, z = 0, (1d)

where w1,2 = ηx,y/√

1 + η2x + η2y . Governing equations (1)then can be nondimensionalized by the introduction of thefollowing scaled variables:

x∗,y∗ = x,yλ

, z∗ = zh

, η∗ = ηa, t∗ = t

√gh

λ,

(2)φ∗ = φh

λa√

gh, T̃ = T

ρgλ2, � = a

h, δ = h

λ,

where a is the characteristic amplitude. For the sake ofnotational simplicity all asterisks will be dropped from thispoint onward.

We are interested in weakly nonlinear harmonic waves ofwave number k with slowly varying amplitudes in both thex and y directions. To achieve this solution, we assume � �O(1) (but leaving δ to be arbitrary) and introduce the followingdifferent-scale variables:

ξ = x − cpt, ζ = �(x − cgt), Y = �y, τ = �2t, (3)where cp(k),cg(k) are, respectively, the phase and groupvelocity of the carrier wave. We further assume that thesolution to the governing equations can be expressed bya convergent asymptotic expansion in terms of the smallparameter �. In terms of new variables [Eq. (3)], we suggest

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http://dx.doi.org/10.1103/PhysRevE.88.035201

BRIEF REPORTS PHYSICAL REVIEW E 88, 035201 (2013)

the form

φ(ξ,ζ,Y,z,τ )

= f0(ζ,Y,τ ) +∞∑

n=0�n

{n+1∑m=0

Fnm(z,ζ,Y,τ )Em + c.c.

}, (4)

η(ξ,ζ,Y,τ ) =∞∑

n=0�n

{n+1∑m=0

Anm(ζ,Y,τ )Em + c.c.

}, (5)

where E = exp (ikξ ) and A00 = 0. At the second order ofperturbation analysis, equations governing the evolution ofthe envelope A01(ζ,Y,τ ) and the mean field f0(ζ,Y,τ ) obtainin the form (see, e.g., Ref. [19] for details)

(1 − c2g

)f0,ζ ζ + f0,YY = −β

2

c2p

(2cpβ

+ cgcosh2 δk

)(|A0|2),ζ

(6a){−2ikcpA0,τ − kcpω′′(k)A0,ζ ζ − cpcgA0,YY

+ k2(2cp + βcgsech2δk)A0f0,ζ + k2

2c2p

A0|A0|2

}= 0,

(6b)

where

= [(10σ 4 − 2σ 6 − 14σ 2 + 6)t̃4+ (40σ 4 − 8σ 6 + 39 − 69σ 2)t̃3+ (−12σ 6 + 69 − 117σ 2 + 63σ 4)t̃2+ (45 − 8σ 6 − 74σ 2 + 46σ 4)t̃ + 9 − 12σ 2+ 13σ 4 − 2σ 6]/[(−3 + σ 2)t̃ + σ 2]. (7)

with t̃ ≡ T̃ k2, σ ≡ tanh δk, c2p = β tanh δk/δk, and β = 1 + t̃ .These equations are, after some algebra, respectively,Eqs. (2.12) and (2.13) of Ref. [19] (note the typo in Eq. (2.12)of Ref. [19] where the “T̃ ” in the numerator inside the bracketis extra. This is corrected in their later expressions such asEq. (2.17)).

To obtain a canonical form of Eqs. (6), we define thefollowing new variables:

ζ ∗ = ζ√

2k

cg∣∣1 − c2g∣∣ , Y ∗ = Y

√2k

cg,

(8)

u∗ =√|ν1 + gν2|

2A0, v

∗ = −ν22

(−f0ζ + g|A0|2),

where

ν1 = k4c3p

,

ν2 = k2cp

(2cp + βcgsech2δk),

g = − 2βν2kcp

(1 − c2g

) .

Upon substitution in Eqs. (6) and after dropping asterisks weobtain

iuτ + spuζζ + uYY − 4ru|u|2 − 2uv = 0, (9a)svζζ − vYY − 4rq|u|2YY = 0, (9b)

in which

p = − ω′′k

cg(1 − c2g

) , q = gν22(ν1 + gν2) ,

(10)r = sign (ν1 + gν2), s = sign

(c2g − 1

).

The Davey-Stewartson equation [Eqs. (9)] is integrableonly in the limiting case of p = q = 1 [28]. In this limit,if s = +1 (s = −1), then Eqs. (9) is an elliptic-hyperbolic(hyperbolic-elliptic) equation and is called the DSI (DSII)equation [30]. For DSI and DSII we have r = +1,−1, andtherefore, analogously to NLS, these equations are calledfocusing DSI and defocusing DSII. DSII is the long wave limitof Eqs. (9) and admits N -soliton, single lump solutions [31],blowup solutions [32], and several other interesting properties[33, §5]. DSI in physical space corresponds to the shallowwater limit and when the group velocity of the carrier wavesexceeds the speed of long waves (cg > 1). This is possibleonly when surface tension is very strong and water depthis very shallow, i.e., when kh �1 and the Bond numberBo ≡ T̃ /δ2 = T/ρgh2 > 1/3. For regular water the lattercondition is satisfied for depths less than 5 mm. The DSIequation is known to admit dromion solutions. In fact, in thislimit of the Davey-Stewartson equation, closed-form dromionsolutions can be obtained by a number of different techniquessuch as utilizing the Bäcklund transformation [26], the inversescattering transform [4], the bilinear-direct method [27,34],and the Wronskian formulation [35].

The objective of this research is to determine if dromionsexist outside the range of validity of DSI, that is, for subcriticalBond numbers (Bo < 1/3) and for deeper waters [kh � O(1)],where the Davey-Stewartson is known to be nonintegrable[28]. For this range, we note that dromions, if they exist, mustbelong to the focusing elliptic-hyperbolic subfamily of theDavey-Stewartson equation, which corresponds to s = +1,r = +1, and p > 0 in Eqs. (9). Therefore the focus of thisarticle will be Eqs. (9) with s = r = +1 but for general p = p(Bo,kh) and q = q (Bo,kh). Clearly, DSI is a subset of thisrange in the limiting case of p = q = 1.

Here we present an iterative numerical scheme that canconverge to dromion solutions for a broad range of Bo andkh. To achieve this we transform Eqs. (9) to the dromionco-moving frame of reference by the change of variablesζ ∗ = ζ − cζ τ and Y ∗ = Y − cY τ , where �vd = cζ ζ̂ + cY Ŷ isthe (a priori unknown) dromion’s velocity. The dromion mustlook stationary in the (ζ ∗,Y ∗) space, except perhaps for a time-dependent phase. A further assumption of a variable phase forthe envelope in the form u(ζ ∗,Y ∗,τ ) = u∗(ζ ∗,Y ∗) exp (iατ )renders the evolution equation for the dromion time indepen-dent. The governing equation for u∗(ζ ∗,Y ∗) after droppingasterisks reads

−αu − icζ uζ − icY uY + puζζ + uYY − 4u|u|2 − 2uv = 0,(11a)

vζζ − vYY − 4q|u|2YY = 0. (11b)

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Starting from initial profiles (u0,v0) Eq. (11a) can be solvedusing a pseudospectral method to achieve an updated value foru. Specifically, since the dromion solution for u must decay tozero at far distances (and if the domain is large enough), Fourierexpansion by the use of the efficient fast-Fourier transform(FFT) algorithm is utilized. In our pseudo-spectral algorithmall linear terms are calculated in spectral domain and nonlinearterms in the physical domain. Equation (11b) is, however, ahyperbolic equation and, if a dromion is expected, its solutionv does not decay to zero at far but to four tracks that extend toinfinity with shapes and directions that are not known a priori.Hyperbolicity of the equation, nonzero tracks at infinity, andthe fact that these tracks, in general, are not periodic makenumerical treatment of this equation challenging [5,36,37]. Amodification of a recently proposed numerical algorithm ([5],for a similar equation in the context of flexural-gravity waves)will be used here to treat Eq. (11b). Specifically, we considerone of the variables, say, ζ , as a pseudotime. As a result,Eq. (11b) can be considered a forced wave equation with aforcing function 4q|u|2YY . Starting from an initial functionv(ζ = 0,Y ), a time-integration algorithm (we use fourth-orderRunge-Kutta) can be used to find v over the entire domain(or, alternatively, by the method of characteristics). Thereforethe problem of solving for v over the entire area reduces tosolving for the initial v(ζ = 0,Y ). This initial profile then canbe corrected using any recursive root-finding algorithm (suchas the Newton-Raphson method) until a proper convergenceis achieved. It turns out that the convergence efficiency ofthe iterative algorithm is significantly higher if the problemis solved for the difference between new dromion and theinitial dromion. Convergence is reached when the absoluteerror (residue of the normalized governing equations whenthe iterated solution is substituted) reaches within computerarithmetic accuracy (1 × 10−16). It is to be noted that theresultant dromion, although qualitatively similar in the shape,may have significantly different scales than the initial dromionand therefore must not be considered as a perturbation to theinitial condition.

To find new dromion solutions to the Davey-Stewartsonequation for the range Bo < 1/3 and kh � O(1) we firstconsider Bo and kh outside but close to the domain of validityof DSI. For such domains the analytical dromion solution ofDSI (e.g., Ref. [27]) proves to be a sufficiently good initialcondition and the algorithm quickly converges. New dromionsolutions are then used as initial conditions (i.e., numericalcontinuation) in further steps to explore farther areas of Boand kh space.

A finite-depth capillary-gravity dromion solution to theDavey-Stewartson equation is shown in Figs. 1(a) and 1(b)for kh = 2 and Bo = 0.4. Dromion solutions can be found foryet deeper waters, as shown in Figs. 1(c) and 1(d), which isfor kh = 9 and Bo = 0.07. For each set of parameters a widerange of dromions can be found depending on the choice ofthe initial condition. While having similar qualitative features,they may differ very much in size. For a given initial conditionand kh, as Bond number increases, the amplitude of u increases(sometimes significantly) and, likewise, v (but much less). Asimilar trend is observed for a fixed Bo as kh increases.

A comprehensive search of the entire Bo-kh parameterspace reveals that capillary-gravity dromions can exist for a

(a) (b)

(c) (d)

FIG. 1. (Color online) Dromion solutions of capillary-gravitywave packets [Eqs. (9)] for the case kh = 2, Bo = 0.4 [(a) and (b)]and kh = 9, Bo = 0.07 [(c) and (d)]. Plotted are, respectively, themagnitude of the amplitude |u| [(a) and (c)] and the negative ofthe mean-current, −v [(b) and (d)]. In the physical domain withwater of surface tension Tw = 0.0728 N/m, the top and bottomfigures, respectively, correspond to water depths of h = 4.3, 10.3 mm,wavelengths of λ = 13.5, 7.2 mm, and peak amplitudes of ηmax = 1.6,0.14 mm.

wide range of water depths and Bond numbers beyond theclassic range of DSI, as seen from the hatched area of Fig. 2. Wefurther show in this figure the nature of the Davey-Stewartsonequation as a function of Bond number and kh. Specifically, fora given kh, as Bond number increases, the Davey-Stewartsonequation changes, over the solid and dash-dotted boundarylines, from hyperbolic-elliptic to elliptic-elliptic and, finally,to elliptic-hyperbolic (cf. Refs. [11,38]). The dashed and

FIG. 2. (Color online) Dromion solutions of capillary-gravitywave packets over the Bo-kh parameter space (hatched area). Solidand dash-dotted lines separate, from left to right, respectively, thehyperbolic-elliptic (HE), elliptic-elliptic (EE), and elliptic-hyperbolic(EH) domains of the Davey-Stewartson equation while dashed anddash-dotted lines separate r = −1 from r = +1 (note that thedash-dotted line is a common line and has two functions). Dromionsare expected only in the focusing (r = +1) EH area of the parameterspace. Top (bottom) [greenish (brownish)] shaded areas show therange of validity of DSI (DSII). Hatched area shows the subset of theparameter space for which we found dromion solutions.

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dash-dotted lines indicate when the unit function r in Eqs. (9)changes sign. The bottom (brownish) shaded area is forkh � 1 and Bo < 1/3 and is therefore associated with DSII.

As elaborated on before, of particular interest is the fo-cusing elliptic-hyperbolic subfamily of the Davey-Stewartsonequation (the area between the right-most dashed line and thedash-dotted line in Fig. 2) for which DSI (top greenish shadedarea) is a subset. For the DSI range in Fig. 2, a closed-formanalytical dromion solution exists [19].

The hatched area in Fig. 2 is the area for which weconsistently were able to converge to a dromion solution (tothe computer accuracy). Clearly, the dromion solution existsfor a broad range of kh � O(1) and Bo < 1/3. For larger kh,however, the range of Bo yielding dromion solutions becomesnarrower and fades away eventually (as is seen in Fig. 2). Thisis expected because as the water depth increases the couplingbetween the surface profile and the mean-current becomesweaker. In this limit, Davey-Stewartson asymptotically ap-proaches the two-dimensional nonlinear Schrödinger equationfor which a dromion solution does not exist.

The convergence efficiency of our computational schemedecreases as we get closer to the boundaries of the focusing EHarea. The hatched area corresponds to dromions that a conven-tional PC can achieve in a finite time, but its boundaries (dottedline in Fig. 2) can be pushed further and more dromions can beobtained using higher-performance computational facilities.

In summary, we showed that dromions can exist forsubcritical Bond numbers (Bo < 1/3) and for finite-depth

waters [kh � O(1)], a range for which the Davey-Stewartsonequation is nonintegrable. Physically speaking, this means thatdromions are not limited to water depths of 5 mm or less (aspredicted by previous theories), but, as we showed, can beexpected in much deeper waters. These new dromions arefound via a pseudospectral scheme and by treating the spatialhyperbolic equation as a wave equation. We also demonstrated,via a comprehensive search of the entire Bo-kh parameterspace, that these structures exist over a large subspace of thefocusing elliptic-hyperbolic Davey-Stewartson equation.

From practical point of view how dromions are generatedand where we should expect to see them are important questionsand subjects of ongoing research (cf. Ref. [8]). The effectof viscosity is expected to be higher for smaller dromionsand results in the attenuation of these structures along thepropagation path, details of which require direct numericalsimulation and are beyond the scope of this paper.

The Davey-Stewartson equation describes physical phe-nomena in several other areas of science, such as quantum fieldtheory [21], ferromagnetism [22], plasma physics [23,24], andnonlinear optics [25]. Results and techniques developed herefor finding new solutions to the Davey-Stewartson equationmay be of interest to the researchers in these fields.

We thank L. A. Couston and P. Hassanzadeh for carefulreading of the manuscript and useful comments. The supportfrom the American Bureau of Shipping is gratefully acknowl-edged.

[1] It is to be noted that two-dimensional harmonic waves cantransport mass, momentum, and energy as a result of their freesurface displacement and horizontal velocity having the samephases.

[2] R. Beals and R. R. Coifman, Physica D 18, 242 (1986).[3] W. Craig, Philos. Trans. R. Soc. London A 360, 2127

(2002).[4] A. Fokas and P. Santini, Physica D 44, 99 (1990).[5] M.-R. Alam, J. Fluid Mech. 719, 1 (2013).[6] K. Berger and P. Milewski, SIAM J Appl. Math 61, 731

(2000).[7] B. Kim and T. R. Akylas, J. Fluid Mech. 540, 337 (2005).[8] J. D. Diorio, Y. Cho, J. H. Duncan, and T. R. Akylas, J. Fluid

Mech. 672, 268 (2011).[9] Y. Cho, J. D. Diorio, T. R. Akylas, and J. H. Duncan, J. Fluid

Mech. 672, 288 (2011).[10] S. Manakov, V. Zakharov, L. Bordag, A. Its, and V. Matveev,

Phys. Lett. A 63, 205 (1977).[11] M. Ablowitz and H. Segur, J. Fluid Mech. 92, 691 (1979).[12] J. Satsuma and M. Ablowitz, J. Math. Phys. 20, 1496

(1979).[13] M. Groves and S. Sun, Arch. Ration. Mech. Anal. 188, 1 (2008).[14] E. I. Parau, J.-M. Vanden-Broeck, and M. J. Cooker, J. Fluid

Mech. 536, 99 (2005).[15] D. Benney and G. Roskes, Stud. Appl. Math. 48, 377 (1969).[16] A. Davey and K. Stewartson, Proc. R. Soc. London A 338, 101

(1974).[17] V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).

[18] H. Hasimoto and H. Ono, J. Phys. Soc. Jpn. 33, 805 (1972).[19] V. Djordjevic and L. Redekopp, J. Fluid Mech. 79, 703

(1977).[20] P. A. Milewski and Z. Wang, Stud. Appl. Math. 131, 135

(2013).[21] C. L. Schultz, M. J. Ablowitz, and D. Bar Yaacov, Phys. Rev.

Lett. 59, 2825 (1987).[22] H. Leblond, J. Phys. A: Math. Gen. 32, 7907 (1999).[23] A. P. Misra, M. Marklund, G. Brodin, and P. K. Shukla, Phys.

Plasmas 18, 042102 (2011).[24] W.-S. Duan, Phys. Plasmas 10, 3022 (2003).[25] H. Leblond, in Soliton-driven Photonics, edited by A. D.

Boardman and A. P. Sukhorukov, NATO Science Series II: Math-ematics, Physics and Chemistry Vol. 31 (Springer, Netherlands,2001), p. 215.

[26] M. Boiti, J.-P. Leon, L. Martina, and F. Pempinelli, Phys. Lett.A 132, 432 (1988).

[27] C. R. Gilson and J. J. C. Nimmo, Proc. R. Soc. London A 435,339 (1991).

[28] E. I. Shul’man, Theor. Math. Phys. 56, 720 (1983).[29] Davey-Stewartson can be derived without the irrotationality

assumption [39–41] (although the derivation is more in-volved) and, hence, results presented in this paper are moregeneral.

[30] A. Fokas and M. Ablowitz, J. Math. Phys. 25, 2494 (1984).[31] V. Arkadiev, A. Pogrebkov, and M. Polivanov, Physica D 36,

189 (1989).[32] T. Ozawa, Proc. R. Soc. A 436, 345 (1992).

035201-4

http://dx.doi.org/10.1016/0167-2789(86)90184-3http://dx.doi.org/10.1098/rsta.2002.1065http://dx.doi.org/10.1098/rsta.2002.1065http://dx.doi.org/10.1016/0167-2789(90)90050-Yhttp://dx.doi.org/10.1017/jfm.2012.590http://dx.doi.org/10.1137/S0036139999356971http://dx.doi.org/10.1137/S0036139999356971http://dx.doi.org/10.1017/S0022112005005823http://dx.doi.org/10.1017/S0022112010005999http://dx.doi.org/10.1017/S0022112010005999http://dx.doi.org/10.1017/S0022112010006002http://dx.doi.org/10.1017/S0022112010006002http://dx.doi.org/10.1016/0375-9601(77)90875-1http://dx.doi.org/10.1017/S0022112079000835http://dx.doi.org/10.1063/1.524208http://dx.doi.org/10.1063/1.524208http://dx.doi.org/10.1007/s00205-007-0085-1http://dx.doi.org/10.1017/S0022112005005136http://dx.doi.org/10.1017/S0022112005005136http://dx.doi.org/10.1098/rspa.1974.0076http://dx.doi.org/10.1098/rspa.1974.0076http://dx.doi.org/10.1007/BF00913182http://dx.doi.org/10.1143/JPSJ.33.805http://dx.doi.org/10.1017/S0022112077000408http://dx.doi.org/10.1017/S0022112077000408http://dx.doi.org/10.1111/sapm.12005http://dx.doi.org/10.1111/sapm.12005http://dx.doi.org/10.1103/PhysRevLett.59.2825http://dx.doi.org/10.1103/PhysRevLett.59.2825http://dx.doi.org/10.1088/0305-4470/32/45/308http://dx.doi.org/10.1063/1.3574913http://dx.doi.org/10.1063/1.3574913http://dx.doi.org/10.1063/1.1581282http://dx.doi.org/10.1016/0375-9601(88)90508-7http://dx.doi.org/10.1016/0375-9601(88)90508-7http://dx.doi.org/10.1098/rspa.1991.0148http://dx.doi.org/10.1098/rspa.1991.0148http://dx.doi.org/10.1007/BF01027548http://dx.doi.org/10.1063/1.526471http://dx.doi.org/10.1016/0167-2789(89)90258-3http://dx.doi.org/10.1016/0167-2789(89)90258-3http://dx.doi.org/10.1098/rspa.1992.0022

BRIEF REPORTS PHYSICAL REVIEW E 88, 035201 (2013)

[33] M. Ablowitz and P. Clarkson, Solitons, Nonlinear EvolutionEquations and Inverse Scattering (Cambridge University Press,Cambridge, 1991).

[34] R. Hirota, The Direct Method in Soliton Theory (CambridgeUniversity Press, Cambridge, 2004).

[35] J. Hietarinta and R. Hirota, Phys. Lett. A 145, 237 (1990).[36] P. White and J. Weideman, Math. Comput. Simul. 37, 469

(1994).[37] C. Besse, N. J. Mauser, and H. P. Stimming, ESAIM: Math.

Model. Num. 38, 1035 (2004).

[38] F. Dias and M. Haragus-Courcelle, Stud. Appl. Math. 104, 91(2000).

[39] T. Colin, F. Dias, and J.-M. Ghidaglia, Eur. J. Mech. B/Fluids14, 775 (1995).

[40] T. Colin, F. Dias, and J.-M. Ghidaglia, in ContemporaryMathematics: Mathematical Problems in the Theory of WaterWaves, edited by F. Dias, J.-M. Ghidaglia, and J.-C. Saut(American Mathematical Society, Providence, Rhode Island,1996), Vol. 200, p. 47.

[41] F. Dias and C. Kharif, Annu. Rev. Fluid Mech. 31, 301 (1999).

035201-5

http://dx.doi.org/10.1016/0375-9601(90)90357-Thttp://dx.doi.org/10.1016/0378-4754(94)00032-8http://dx.doi.org/10.1016/0378-4754(94)00032-8http://dx.doi.org/10.1051/m2an:2004049http://dx.doi.org/10.1051/m2an:2004049http://dx.doi.org/10.1111/1467-9590.00132http://dx.doi.org/10.1111/1467-9590.00132http://dx.doi.org/10.1146/annurev.fluid.31.1.301